% ELECTRIC VECTOR POTENTIAL
Consider Maxwell's equations (\ref{eqn:gefl})--(\ref{eqn:al}) in
an electric source free region, $q_{ev}=\vec{J}_i=0$.
\begin{align}
    \nabla\cdot\varepsilon\vec{E}_F&=0\label{eqn:geflF}\\
    \nabla\cdot\mu\vec{H}_F&=q_{mv}\label{eqn:gmflF}\\
    \nabla\times\vec{E}_F&=-j\omega\mu'\vec{H}_F-\vec{M}_i\label{eqn:flF}\\
    \nabla\times\vec{H}_F&=j\omega\varepsilon'\vec{E}_F\label{eqn:alF}
\end{align}
The vector identity, $\nabla\cdot\bigl(\nabla\times\vec F\bigr)=0$
, can be used to define an electric vector potential, $\vec F$.
Multiplying this identity by the constant $-\varepsilon$ and
setting it equal to (\ref{eqn:geflF}) yields
$$
    \nabla\cdot\varepsilon\vec{E}_F
    =\nabla\cdot\bigl(-\nabla\times{\varepsilon}\vec{F}\bigr)
    =0
$$
and solving for $\vec E_F$
\begin{equation}\label{eqn:EsubF}
    \vec{E}_F
    =-\nabla\times\vec{F}
\end{equation}
The subscript F in is used as a reminder that the electric field
was solved from an electric vector potential. The units of
$\vec{F}$ is volts. Substituting (\ref{eqn:EsubF}) into Ampere's
law (\ref{eqn:alF}) yields,
\begin{equation}\label{eqn:1.22sub1.21}
    \nabla\times\vec{H}_F
    =-j\omega\varepsilon'\nabla\times\vec{F}
\end{equation}
Using the vector identity, $\nabla\times\bigl(-\nabla\Phi\bigr)=0$
where $\Phi$ is any arbitrary scalar, and rearranging yields,
(\ref{eqn:1.22sub1.21})
$$
    \nabla\times\left[\vec{H}_F+j\omega\varepsilon'\vec{F}\right]
    =\nabla\times\bigl(-\nabla\Phi_m\bigr)=0
$$
Solving for $\vec{H}_F$ yields a magnetic field due to a magnetic
scalar potential, $\Phi_m$ and electric vector potential, $\vec
F$.
\begin{equation}\label{eqn:HsubF'}
    \vec{H_F}
    =-\nabla\Phi_m-j\omega\varepsilon'\vec{F}
\end{equation}
Taking the curl of both sides of (\ref{eqn:EsubF}) and using the
vector identity
$\nabla\times\nabla\times\vec{F}=\nabla\bigl(\nabla\cdot\vec{F}\bigr)-\nabla^2\vec{F}$
\begin{equation}\label{eqn:t1F}
    -\bigl(\nabla\times\vec E_F\bigr)
    =\nabla\times\nabla\times\vec{F}
    =\nabla\bigl(\nabla\cdot\vec F\bigr)-\nabla^2\vec{F}
\end{equation}
Equating (\ref{eqn:t1F}) with Faraday's law (\ref{eqn:flF}) yields
\begin{equation}\label{eqn:t2F}
j\omega\mu'\vec{H}_F+\vec{M}_i
=\nabla\bigl(\nabla\cdot\vec{F}\bigr)-\nabla^2\vec{F}
\end{equation}
Substituting (\ref{eqn:HsubF'}) into (\ref{eqn:t2F}) and
rearranging yields the following non homogeneous wave equation
\begin{equation}\label{eqn:wave0F}
\nabla^2\vec{F}+\omega^2\mu'\varepsilon'\vec{F}
=-\vec{M}_i+\nabla\bigl(\nabla\cdot\vec{F}+j\omega\mu'\Phi_m\bigr)
\end{equation}
Since the divergence has not yet been specified, it can be chosen
to be
\begin{equation}\label{eqn:lgf}
    \nabla\cdot\vec{F}=-j\omega\mu'\Phi_m
\end{equation}
This is called the Lorentz gauge. $\Phi_m$ is now found to be
\begin{equation}\label{eqn:Phi_m}
    \Phi_m
    =-\frac{\nabla\cdot\vec{F}}{j\omega\mu'}
\end{equation}
Equation (\ref{eqn:wave0F}) now reduces to,
\begin{equation}\label{eqn:waveF}
    \nabla^2\vec{F}-\gamma^2\vec{F}
    =\nabla^2\vec{F}+k^2\vec{F}
    =-\vec{M}_i
\end{equation}
where,
$$
    \gamma^2
    =-k^2
    =-\omega^2\mu'\varepsilon'\eqno(\ref{eqn:gk})
$$
Equation (\ref{eqn:waveF}) is recognized as a non-homogeneous
vector wave equation and (\ref{eqn:gk}) is a complex propagation
constant. Finally, substituting (\ref{eqn:Phi_m}) into
(\ref{eqn:HsubF'}) yields a magnetic fields due to and electric
vector potential,
\begin{align}
    \vec{H}_F&=\frac{\nabla\bigl(\nabla\cdot\vec{F}\bigr)}{j\omega\mu'}-j\omega\varepsilon'\vec{F}\label{eqn:H_F1}\\
    &=\frac{1}{j\omega\mu'}\biggl[\nabla\bigl(\nabla\cdot\vec{F}\bigr)-(\gamma^2=-k^2)\vec{F}\biggr]\label{eqn:H_F2}\\
    &=\frac{1}{j\omega\mu'}\biggl[\nabla\bigl(\nabla\cdot\vec{F}\bigr)-\nabla^2\vec{F}-\vec{M}_i\biggr]\label{eqn:H_F3}\\
    &=\frac{1}{j\omega\mu'}\biggl[\nabla\times\nabla\times\vec{F}-\vec{M}_i\biggr]\label{eqn:H_F4}
\end{align}
